Equivalence Class in Congruent Modulo

Can you the think the set of remainders as a class?

See the example. Let a == b mod 2. Every integer is congruent modulo 2 to exactly either 0 or 1. For example, 17 when divided by 2 will give you remainder 1 and 24 when divided by 2 will give you remainder 0. In fact, every integer when divided by 2 will give you remainder either 0 or 1. Hence all integers on number line shall be grouped in two classes of numbers wrapped up around 0 or 1.

These classes are known equivalence classes or residue classes.

In general, any modulo n has n residue classes, one for each integer from 0 to n-1.

Every integer is congruent modulo 4 to exactly 0, 1, 2 and 3. Means, when an integer is divided by 4, it shall leave remainder either 0 or 1 or 2 or 3. Hence every integer modulo 4 shall belong to some member of this equivalence class {0,1,2,3}.

Hence, each and every congruent, say y, belonging to the set {0, 1, 2, 3, 4,, ……, (n-2), (n-1)} can be thought of a box containing an infinite numbers, say x, satisfying only one criteria, i.e. the number x when divided by n leaves y as remainder.